Editing Infinite impulse response (section)

=== Impulse Invariance ===
Impulse invariance is a technique for designing discrete-time infinite-impulse-response (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of the discrete-time system. 
Impulse invariance is one of the commonly used methods to meet the two basic requirements of the mapping from the s-plane to the z-plane. This is obtained by solving the T(z) that has the same output value at the same sampling time as the analog filter, and it is only applicable when the inputs are in a pulse.<br />
Note that all inputs of the digital filter generated by this method are approximate values, except for pulse inputs that are very accurate. This is the simplest IIR filter design method. It is the most accurate at low frequencies, so it is usually used in low-pass filters.

For Laplace transform or z-transform, the output after the transformation is just the input multiplied by the corresponding transformation function, T(s) or T(z). Y(s) and Y(z) are the converted output of input X(s) and input X(z), respectively.<br />
:<math>Y(s)=T(s)X(s)</math><br />
:<math>Y(z)=T(z)X(z)</math><br />
When applying the Laplace transform or z-transform on the unit impulse, the result is 1. Hence, the output results after the conversion are <br />
:<math>Y(s)=T(s)</math><br />
:<math>Y(z)=T(z)</math><br />
Now the output of the analog filter is just the inverse Laplace transform in the time domain.<br />
:<math>y(t)=L^{-1}[Y(s)]=L^{-1}[T(s)]</math><br />
If we use nT instead of t, we can get the output y(nT) derived from the pulse at the sampling time. It can also be expressed as y(n)<br />
:<math>y(n)=y(nT)=y(t)|_{t=sT}</math><br />
This discrete time signal can be applied z-transform to get T(z)<br />
:<math>T(z)=Y(z)=Z[y(n)]</math><br />
:<math>T(z)=Z[y(n)]=Z[y(nT)]</math><br />
:<math>T(z)=Z\left\{L^{-1}[T(s)]_{t=nT}\right\}</math><br />
The last equation mathematically describes that a digital IIR filter is to perform z-transform on the analog signal that has been sampled and converted to T(s) by Laplace, which is usually simplified to<br />
:<math>T(z)=Z[T(s)]*T</math><br />
Pay attention to the fact that there is a multiplier T appearing in the formula. This is because even if the Laplace transform and z-transform for the unit pulse are 1, the pulse itself is not necessarily the same. For analog signals, the pulse has an infinite value but the area is 1 at t=0, but it is 1 at the discrete-time pulse t=0, so the existence of a multiplier T is required.<br />